Method and system for simulating risk factors in parametric models using risk neutral historical bootstrapping

ABSTRACT

An improved method for simulating noise-varying risk factor values in a parametric simulation comprises analyzing historical data to determine the actual value of the risk factors and other attributes in the model and using this data to generate historical residual values which reproduces the historical price when used in the model with corresponding historical attribute values. The set of historical residual values is standardized and can be bootstrapped to increase the number of members in the set or vary the sets properties. Values of the historical residuals are then selected, e.g., at random, and used in place of the random noise components to produce simulated risk factor values which are used in the parametric model to simulate the evolution of the instrument price.

RELATED APPLICATIONS

The present application is a continuation application of priorapplication Ser. No. 09/896,660 filed Jun. 29, 2001 to which priorityunder 35 USC §120 is claimed and is hereby incorporated by reference.

FIELD OF THE INVENTION

This invention is related to a method and system for measuring marketand credit risk and, more particularly, to an improved method for thesimulation of risk factors in parametric models for use in making valueat risk and other risk evaluations.

BACKGROUND

A significant consideration which must be faced by financialinstitutions (and individual investors) is the potential risk of futurelosses which is inherent in a given financial position, such as aportfolio. There are various ways for measuring potential future riskwhich are used under different circumstances. One commonly acceptedmeasure of risk is the value at risk (“VAR”) of a particular financialportfolio. The VAR of a portfolio indicates the portfolio's market riskat a given percentile. In other words, the VAR is the greatest possibleloss that the institution may expect in the portfolio in question with acertain given degree of probability during a certain future period oftime. For example, a VAR equal to the loss at the 99^(th) percentile ofconfidence level indicates that there is only a 1% chance that the losswill be greater than the VAR during the time frame of interest.

Generally, financial institutions maintain a certain percentage of theVAR in reserve as a contingency to cover possible losses in theportfolio in a predetermined upcoming time period. It is important thatthe VAR estimate be accurate. If an estimate of the VAR is too low,there is a possibility that insufficient funds will be available tocover losses in a worst-case scenario. Overestimating the VAR is alsoundesirable because funds set aside to cover the VAR are not availablefor other uses.

To determine the VAR for a portfolio, one or more models whichincorporate various risk factors are used to simulate the price of eachinstrument in the portfolio a large number of times using an appropriatemodel. The model characterizes the price of the instrument on the basisof one or more risk factors, which can be broadly considered to be amarket factor which is derived from tradable instruments and which canbe used to predict or simulate the changes in price of a giveninstrument. The risk factors used in a given model are dependent on thetype of financial instrument at issue and the complexity of the model.Typical risk factors include implied volatilities, prices of underlyingstocks, discount rates, loan rates, and foreign exchange rates.Simulation involves varying the value of the risk factors in a model andthen using the model to calculate instrument prices in accordance withthe selected risk factor values. The resulting price distributions areaggregated to produce a value distribution for the portfolio. The VARfor the portfolio is determined by analyzing this distribution.

There are two alternative simulation techniques which are conventionallyused during risk analysis, such as VAR calculations: parametricsimulation and historical simulation.

In a parametric simulation, the change in value of a given price for asecurity is simulated by changing the value of the risk factors in themodel from their initial values according to a stochastic or randomfunction. A well known model used in option pricing is the Black-Scholesmodel which models the change in a stock price S over a time interval tas a function of σ √{square root over (Δ/ε)}, where σ is a risk factorindicating the volatility of the price, and ε is a random component.Parametric simulation has the advantage of being very flexible. Forexample, the values of the parameters which define the model can beadjusted as required to make the model risk neutral. In addition, whenthe starting values of the model parameters cannot be determined orimplied from actual data, default parameters can be used until reliablehistorical or market data is available.

A serious drawback to this technique, however, is that the noisecomponents ε used to vary the risk factor values are generally assumedto have a normal distribution. In reality, low probability events occurwith more frequency than in a normal distribution. As a result,so-called “fat-tails” of the probability curve must be explicitlydefined in the model and used to alter the normal distribution of ε.

Another problem with parametric models is that the model must expresslymodel cross-correlations between various risk factors. Typically, avariance-covariance matrix is used to preserve a predeterminedcorrelation between the various risk factors during a simulation. Anunderlying assumption to this technique is that the correlations betweenvarious factors are constant across the range of input parameters.However, the correlations can vary depending on the circumstances.Detecting these variations and compensating for them through the use ofmultiple variance-covariance matrices is difficult and can greatlycomplicate the modeling process. In addition, the computational cost ofdetermining the cross-correlations grows quadratically with the numberof risk factors. It is not unusual for large derivative portfolios todepend on 1000 or more risk factors and determining thecross-correlations for the risk factors quickly becomes unmanageable,particularly when the simulation process must be run daily.

An alternative to parametric simulation is historical simulation. In ahistorical simulation, a historical record of data is analyzed todetermine the actual risk factor values. To simulate price evolution,risk factor values are selected at random from the historical set andapplied to the model to determine the next price in the simulation. Thisapproach is extremely simple. Because historical data is used as adirect source for the risk factor values, the methodology does notrequire calculation of model parameters, such as correlations andvolatilities. Moreover, the fat-tail event distribution and stochasticcorrelations between various factors is automatically reproduced.However, this method is limited because the statistical distribution ofvalues is restricted to the specific historical sequence which occurred.In addition, historical data may be missing or non-existent,particularly for newly developed risk factors, and the historicalsimulation is generally not risk neutral.

Accordingly, there is a need for an improved technique for adjusting thevalue of risk factors during simulation of a financial instrument, e.g.,for use in risk analysis.

SUMMARY OF THE INVENTION

This and other needs are addressed by present invention which providesan improved method for varying the value of risk factors in a parametricsimulation. The new method accurately accounts for “fat-tail”probability distributions and cross-correlation between various riskfactors while allowing the model to be risk neutral. In addition, themethod is suitable for use in developing models which are accurate forboth short horizon VAR simulations and long horizon potential creditexposure (“PE”) simulations.

Initially, the risk factors used in a parametric model of a givenattribute, such as the price of an instrument, are identified. Thehistorical data is analyzed to determine the actual value of those riskfactors over a certain period along with the value of the modeledattribute. The historical risk factor values and the correspondingattribute values are then applied to the parametric model and the modelis solved to derive a set of historical residuals that quantify thevalues of the noise factors ε needed to reproduce the historical valuesof the attribute using the historical risk factor values in the model.The distribution of the residuals values is then standardized. Abootstrapping procedure can be performed to increase the number ofresidual values in the set.

During simulation, values from the standardized set of historicalresiduals are selected at random and used as the ε values in theparametric model. Advantageously, the historically derived residualsretain the underlying correlation between the various risk factors whilestill permitting the model to be risk neutral. As a result, there is noneed to separately determine cross-correlations or correct for themusing a variance-covariance matrix. Further, because the data ishistorically derived, the distribution of residual values retainshistorical fat-tail distributions which are absent in the normaldistributions used in conventional parametric simulation.

BRIEF DESCRIPTION OF THE FIGURES

The foregoing and other features of the present invention will be morereadily apparent from the following detailed description and drawings ofillustrative embodiments of the invention in which:

FIG. 1 is a flow diagram of a process for determining the value at riskfor a portfolio;

FIG. 2 is a diagram indicating the use of matrices of precalculatedsimulated risk factor values to a pricing model to generate a matrix ofsimulated instrument prices; and

FIG. 3 is a flowchart of a method for simulating the risk factor valuesin a financial model in accordance with the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Turning to FIG. 1, there is shown a general flow diagram of a system forevaluating an integrated risk in a portfolio. Initially, the portfolioor position at interest is analyzed to determine the appropriate riskfactors to apply. The risk factors are simulated using a marketsimulation model or other stress scenarios to produce sets of simulatedrisk factors for the time period of interest. The simulated risk factorsare applied to pricing models for the various instruments in theportfolio and a set simulated prices for each of the instruments isgenerated. The simulated values for the priced instruments are thenaggregated to produce a set of simulated prices for the portfolio. Theseprices can then be analyzed to evaluate various risk measures, such asVAR.

According to the invention, a risk neutral bootstrap method is used togenerate a set of historical residuals for a given factor in aparametric model based. The residuals are determined by calculating thevalue of the factor at the time of each historical sample and thendetermining what residual “random” value of ε would be required toproduce the actual historical price or other modeled value using themodel. The distribution of historical residuals is standardized and thenused to generate the simulated factor values which can then be appliedto the model to simulate the performance of the instrument over a giventime period, e.g., in order to determine the VAR.

In an arbitrary parametric model used in simulation, the change in valuedF of a given value F, such as price of a security or a market variableeffecting the price of a variable that is subjected to a “noisy”variation over time, is represented by a model M which includes at leastone parameter a. The variations in factors are induced by random erroror noise terms ε₁ . . . ε_(k):dF=M(F, a ₁ , a ₂ , . . . , a _(n), ε₁, ε₂, ε_(k))  (Equ. 1)The risk factors used in a given model are dependent on the type offinancial instrument at issue. Typical risk factors for the price of asecurity include implied volatilities, prices of underlying stocks,discount rates, loan rates, and financial exchange rates. Some of theserisk factors, such as a time until expiration, are deterministic andhave values which can easily be determined for simulation of futureevents. Other risk factors, however, are dependent on noise-varyingparameters and must therefore be modeled.

A typical model represents one attribute and contains a single noiseterm ε. For example, a well known model of the change in a stock price Sover a time Δt is provided by Black-Scholes as:

$\begin{matrix}{\frac{\Delta\; S}{S} = {{\mu\;\Delta\; t} + {\sigma\sqrt{\Delta\; t\; ɛ}}}} & \left( {{Equ}.\mspace{14mu} 2} \right)\end{matrix}$where μ is a drift parameter, σ is a factor indicating the volatility ofthe price, and ε is a noise component which is used to alter thevolatility by a given amount at each step of the simulation and has avalue between zero and one. For simplicity, this aspect of the inventionwill be illustrated with respect to the Black-Scholes model of Equ. 2.However, the invention can be applied to other, more complex multiplefactor models.

According to the invention, historical data is analyzed to determine theactual value of the factors for each time index i (such as at the end ofeach trading day). This information and the historical values of themodeled value are applied to the model to determine the correspondingresidual values ε_(i) which would recreate the historical value from themodel using the historical factor values. In other words, the arbitrarymodel is solved for each residual value:ε_(i)=M⁻¹(dF, F, a ₁ , a ₂ , . . . , a _(n))  (Equ. 3)Performing this using the example Black-Scholes model of Equ. 2provides:

$\begin{matrix}{ɛ_{i} = {\frac{1}{\sigma_{{imp}_{i}}\sqrt{\Delta\; t}}\left( {\frac{\Delta\; S_{i}}{S_{i}} - {\mu_{{imp}_{i}}\Delta\; t}} \right)}} & \left( {{Equ}.\mspace{14mu} 4} \right)\end{matrix}$where σ_(imp,i) is the implied volatility, μ_(imp,i) is the implieddrift, and ε_(i) is the historical residual value at index i (e.g., timet). Both σ_(imp,i) and μ_(imp,i) can be determined, estimated, orimplied from historical market data using conventional techniques.

The result of this process is a collection of one-dimensional indexedseries of determined historical residual values ε_(i) for a set of timest in the historical record. This series is stored and used as discussedbelow.

For an arbitrary model with k different noise component residuals, therewill be a corresponding series for each residual ε_(1,i) . . . ε_(k,i).As will be appreciated, if multiple residuals are present, there must asufficient number of modeled equations on each historical day to permitthe model to be solved for the residuals at each historical point. Thisis generally the case since some of the noise factors can be solved forusing independent models. For example, a model of the change in thestock of a software company could include risk factors based uponmeasures of the software sector, and the market as a whole, e.g.,ΔS/S=xε₁+y(sector)ε₂+z(market)ε₃. Although this simplistic model hasthree noise factors ε₁ . . . ε₃, the residual values for the sector andmarket could be solved for using separate models of these factors andthen the results used when solving for the remaining value of ε₁.Alternatively, the model can be adjusted, e.g., by combining the effectof various factors, to reduce the number of ε values to a solvablelevel. Appropriate techniques for doing this will be known to those ofskill in the art.

For each of the parameters a₁ . . . a_(n) which were derived from thehistorical data, such as implied drift and volatility, this process alsogenerates a corresponding indexed series of parameter values.Preferably, the sets of implied historical parameters are also storedfor future use. Advantageously, because the historical residual valuesand parameter values are derived from historical data, subsequentanalyses using “new” historical data can re-use much of the previouslycalculated data. The sets of implied parameters and derived residualscan be updated incrementally based on recent historical data points (andthe oldest values deleted as appropriate). This reuse can substantiallyreduce the amount of processing which is required for subsequentsimulations.

The distribution of values in the sequence of determined historicalvalues E, will generally not be standard. According to one aspect of theinvention, the generated historical residual values are used in aparametric model in place of the random components. Thus, the values ofthe historical residuals should be standardized to the range suitablefor the corresponding random component in the model, typically anempirical average E[ε]=0 and a variance var[ε]=1. To preservecorrelations which may exist between different sets of residuals fromthe historical sample, a linear standardization process can be appliedto each residual value series, e.g., ε_(i), to provide a correspondingstandardized series:ε′_(i)=αε_(i)+β  (Equ. 5)where the values of α and β are selected to provide E[ε_(i)′]=0 andvar[ε_(i)′]=1 for the given series of ε_(i) at issue (and may bedifferent for different series).

Finally, the residuals (and determined parameters) are applied to thecorresponding model. Initially, the values of the model parameters, suchas drift and volatility, are determined for the starting point of themodel, such as the most current values. At each simulated step in ascenario run, an index value x is selected at random and the value forthe historical residual for that index value is substituted into themodel to generate a simulated risk factor value which can then be usedin the model to generate a simulated price or other modeled attribute.Preferably, the initial value for the parameters is used to generateeach simulated risk factor value although, in an alternative embodiment,these values could also be adjusted as appropriate.

Thus, for the Black-Scholes model discussed above, the initial values ofσ_(imp) and μ_(imp) to use in the simulation are determined from thestarting conditions. Then, the simulated scenario progresses usingvalues of i selected randomly to access a “random” value of ε_(i)′ fromthe set of historical residuals to produce a simulated valueσ_(imp)ε′_(i) for the volatility risk factor which is used in the pricemodel:

$\begin{matrix}{\frac{\Delta\; S}{S} = {{\mu_{imp}\;\Delta\; t} + {\sigma_{imp}\sqrt{\Delta\; t}ɛ_{i}^{\prime}}}} & \left( {{Equ}.\mspace{14mu} 6} \right)\end{matrix}$More generally, for a randomly selected index value x, the simulationprogresses as:dF=M(F, a ₁ , a ₂ , . . . , a _(n), ε_(1,x), ε_(2,x), ε_(k,x))  (Equ. 7)

In a preferred embodiment, prior to performing the simulation process, abootstrapping process is performed on the historical residuals. Thisprocess can be used to account for certain deficiencies in the data,adjust the statistical distribution, increase the number of availablesamples, or a combination of these or other factors. Variousbootstrapping processes are discussed below.

In some situations, historical data may be missing, incorrect, ornon-existent. This can occur, for example, when the performance of a newsecurity must be simulated. In order to compensate for such gaps, themissing historical data can be back-filled with “synthetic” datagenerated using conventional techniques, such as extrapolating fromvalid data, drawing historical data from similar securities, or applyingperformance models. The synthetic historical data can be replaced byactual historical data as it becomes available. In addition, badhistorical days, such as holidays, can be identified and correspondinghistorical residual values excluded from the set. Similarly, outlierscan also be excluded, such as data points which differ from the mean bymore than a selected multiple of the standard deviation, for example,5.5*sigma.

During a simulation with a large number of scenarios, the number ofhistorical residuals used will typically greatly exceed the actualnumber of samples calculated directly from the historical data. Thus, itmay be necessary to increase the total number of historical residualswhich are available. To address this situation, an n-day bootstrapprocedure can be used to generate additional residual values for useduring simulation. A preferred bootstrapping technique is to sum a set nof randomly selected samples and divide by the square-root of n toproduce a new residual value:

$\begin{matrix}{ɛ^{''} = \frac{\sum\limits_{j = 1}^{n}\; ɛ_{j}^{\prime}}{\sqrt{n}}} & \left( {{Equ}.\mspace{14mu} 8} \right)\end{matrix}$This increases the total number of samples by a power of n (at the costof reducing kurtosis, the fourth moment of the statistical distribution,for higher values of n). Preferably, a two-day bootstrapping is used.For a 250 day history, this process produces a sequence of up to250*250=62,500 samples to draw on. Moreover, the low value of n=2 doesnot significantly reduce any fat-tail which may be present in thedistribution.

In certain circumstances, it may be desirable to provide a sets ofhistorical residuals in which the distribution has been normalized(e.g., the fat-tail has been removed), but the correlations aremaintained. In accordance with the central-limit theorem, as n isincreased, the distribution of the resulting residuals moves moretowards a normal distribution. As a result, using a relatively highvalue of n, such as 6 or more, will artificially remove some or all of adistribution fat-tail which may be present while preserving thecross-correlations. By selecting intermediate values of n, the effect ofthe fat-tail can be reduced without completely eliminating it. Inaddition, by comparing historical distributions with and without thefat-tail it is possible to determine the shape of the fat-tail relativeto a normal distribution.

According to another bootstrapping procedure, the distribution ofresiduals is symmetrized. This is useful for situations where thehistorical data produces variations in a risk factor which are generallyskewed. A symmetrized set can be generated by randomly selecting tworesidual values i and j and combining them as:

$\begin{matrix}{ɛ^{''} = \frac{ɛ_{i}^{\prime} - ɛ_{j}^{\prime}}{\sqrt{2}}} & \left( {{Equ}.\mspace{14mu} 9} \right)\end{matrix}$

Various other bootstrapping techniques known to those of skill in theart can also be used and more than one modification to the originallyderived set of historical residuals can be performed prior to thesimulation. In order to preserve correlations that exist between thevarious sequences of (standardized) historical residuals, the samebootstrapping process should be applied to each historical residualsequence to be used in a simulation to provide new bootstrappedsequences. Preferably, standardization is performed prior to thebootstrapping procedure. However, it is possible to performstandardization after the bootstrapping process.

Regardless of the particular bootstrapping techniques which are used,after the working sets of historical residuals have been generated, theycan be applied to the model to produce simulated instrument prices orother simulated values. In one embodiment, the values of the variousrisk factors used in the model at each set of the simulation can bedetermined “on-the-fly” as the simulation progresses. While this is asuitable process for simulation of a single instrument, when multipleinstruments are simulated, on-the-fly risk factor evaluation may not beefficient because the same risk factor (and set of historical residuals)can be used during the simulation of several different instruments.

Accordingly, in a preferred implementation, and with reference to FIGS.1 and 2, the evolving values of the risk factors themselves aresimulated and the results for each risk factor are stored in acorresponding simulated risk factor matrix. After all of the relevantrisk factors have been simulated and the matrices stored, the simulatedrisk factor data is applied as to the appropriate simulation model toproduce a matrix of simulated price scenarios for the particularinstruments of interest.

When multiple instruments are simulated the simulated price matrices forthose instruments are aggregated using conventional techniques toproduce a matrix of simulated prices for the entire portfolio. Ifmultiple portfolios are being analyzed, each will generally have its ownsimulated pricing matrix which can be used to determine the VAR for theportfolio at a given percentile or for other purposes.

The simulation process as it applies to a single instrument issummarized in the flowchart of FIG. 3. Initially a suitable parametricsimulation model is provided. (Step 30) Next, the values of theparameters for a set of historical data is determined (step 31) and acorresponding sequence of historical residual values are created which,when applied to the model using the historical parameter values,recreate the historical performance of the modeled attribute, such asprice or a noisy factor to be used in a pricing model. (Step 32) Thehistorical residuals are standardized (step 33) and then bootstrapped(step 34).

The working set of historical residuals can then be used to generate asimulated value matrix for each noisy risk factor, each of whichcontains the simulated value of the respective risk factor for each stepof a number of simulated scenarios. (Step 35) Finally, the simulatedrisk factors are applied to the model to generate a corresponding matrixof simulated prices for the matrix which can be used in subsequent riskanalysis. (Step 36)

Through the use of this method, the advantages of a parametricsimulation methodology can be leveraged without having to determine andmodel cross-correlations between risk factors or adjust the random riskfactor variations to reflect a fat-tail distribution. As a furtheradvantage, the present invention provides a mechanism through which asingle model can be developed which accurately models both short-horizonevents (to determine VAR) and long-horizon events (to determine PE, thepotential exposure), as opposed to the conventional practice whichutilizes different models for short and long horizon simulations.

The present invention can be implemented using various techniques. Apreferred method of implementation uses a set of appropriate softwareroutines which are configured to perform the various method steps on ahigh-power computing platform. The input data, the generatedintermediate values, simulated risk factors, priced instruments, andportfolio matrices can be stored in an appropriate data storage area,which can include both short-term (fast access) memory and long-termstorage, for subsequent use. Appropriate programming techniques will beknown to those of skill in the art and the particular techniques useddepend upon implementation details, such as the specific computing andoperating system at issue and the anticipated volume of processing. In aparticular implementation, a Sun OS computing system is used. Thevarious steps of the simulation method are implemented as C++ classesand the intermediate data and various matrices are stored in respectivefiles and databases.

While the invention has been particularly shown and described withreference to preferred embodiments thereof, it will be understood bythose skilled in the art that various changes in form and details can bemade without departing from the spirit and scope of the invention.

1. A financial instrument price simulation processor-implemented methodcomprising: providing via a processor a parametric model for an at leastone financial instrument price having at least one noise-varyingparameter α_(n) with a corresponding noise component ε_(n); determiningvalues for the at least one parameter and the financial instrument priceat various time indices i using historical data; deriving a set ofhistorical residual values ε_(n,i) for each noise component ε_(n), thehistorical residual value ε_(n,i) at index i, when applied to the modelwith the determined parameter values at index i, at least substantiallyreproducing the determined financial instrument price at index i;standardizing each set of historical residual values ε_(n); and usingvalues selected from the set of standardized historical residual valuesε_(n) as the noise component during a simulation of the financialinstrument price via the model, comprising generating a set of simulatedparameter values using values selected from the corresponding set ofstandardized historical residual values ε_(n) for each noise-varyingparameter α_(n) in the model; storing the simulated values for eachparameter; and applying stored simulated parameter values to the modelto produce simulated financial instrument prices.
 2. The method of claim1, wherein the use of values comprises using values for a plurality ofscenarios for each instrument in a portfolio to produce a correspondingset of simulated financial instrument price scenarios for eachinstrument in the portfolio.
 3. The method of claim 2, furthercomprising: aggregating the simulated financial instrument prices toproduce a set of simulated portfolio value scenarios; and analyzing thesimulated portfolio value scenarios to determine a value at risk for theportfolio.
 4. The method of claim 1, further comprising: symmetrizingthe set of historical residual values prior to the use of selectedhistorical residual values.
 5. The method of claim 1, furthercomprising: increasing the quantity of historical residual valuesavailable for use by applying a multi-day bootstrapping procedure to theset of historical residual values.
 6. The method of claim 5, wherein atwo-day bootstrapping procedure is used.
 7. A financial instrumentportfolio attribute value simulation processor-implemented methodcomprising: providing via a processor an at least one parametric modelfor each of a plurality of financial instrument attribute valuesassociated with a financial instrument portfolio, wherein each financialinstrument attribute has at least one noise-varying parameter α_(n) witha corresponding noise component ε_(n); determining values for each ofthe at least one parameters and the financial instrument attributes atvarious time indices i using historical data; deriving a set ofhistorical residual values ε_(n,i) for each noise component ε_(n), thehistorical residual value ε_(n,i) at index i, when applied to the modelwith the determined parameter values at index i, at least substantiallyreproducing the determined financial instrument attribute value at indexi; standardizing each set of historical residual values ε_(n); usingvalues selected from the set of standardized historical residual valuesε_(n) as the noise component during a simulation of the financialinstruments attributes values via the at least one model, wherein theuse of values comprises using values for a plurality of scenarios foreach financial instrument in the portfolio to produce a correspondingset of simulated financial instrument attribute value scenarios for eachinstrument in the portfolio; aggregating the simulated financialinstrument attribute values to produce a set of simulated portfolioattribute value scenarios; and analyzing the simulated portfolioattribute value scenarios to determine an at least one attribute valueat risk for the portfolio.
 8. A financial instrument price simulationprocessor-implemented method comprising: providing via a processor aparametric pricing model having at least one parameter α_(n) with acorresponding noise component ε_(n); determining values for theparameters at various time indices i using historical data; deriving aset of historical residual values ε_(n,i) for each noise componentε_(n), the historical residual value ε_(n,i) at index i, when applied tothe model with the determined parameter values at index i, at leastsubstantially reproducing the attribute price at index i; standardizingeach set of historical residual values ε_(n); applying a multi-daybootstrapping procedure to each set of historical residual values toincrease the quantity of historical residual values in each set; andusing values selected from the set of historical residual values as thenoise component for the corresponding parameter during a simulation ofthe instrument price.
 9. The method of claim 8, wherein the at least oneparameter comprises volatility.
 10. The method of claim 8, wherein theat least one parameter comprises drift.
 11. The method of claim 8,wherein the historical data includes implied drift.
 12. The method ofclaim 8, wherein the historical data includes volatility.
 13. Afinancial instrument price simulation system, comprising: a memory; aprocessor disposed in communication with said memory, and configured toissue a plurality of processing instructions stored in the memory,wherein the processor issues instructions to: provide a parametric modelfor an at least one financial instrument price having at least onenoise-varying parameter α_(n) with a corresponding noise componentε_(n); determine values for the at least one parameter and the financialinstrument price at various time indices i using historical data; derivea set of historical residual values ε_(n,i) for each noise componentε_(n), the historical residual value ε_(n,i) at index i, when applied tothe model with the determined parameter values at index i, at leastsubstantially reproducing the determined financial instrument price atindex i; standardize each set of historical residual values ε_(n); anduse values selected from the set of standardized historical residualvalues ε_(n) as the noise component during a simulation of the financialinstrument price via the model, wherein the processor issues furtherinstructions to: generate a set of simulated parameter values usingvalues selected from the corresponding set of standardized historicalresidual values ε_(n) for each noise-varying parameter α_(n) in themodel; store the simulated values for each parameter; and apply storedsimulated parameter values to the model to produce simulated financialinstrument prices.
 14. The system of claim 13, wherein the processorissues further instructions to: use values for a plurality of scenariosfor each instrument in a portfolio to produce a corresponding set ofsimulated financial instrument price scenarios for each instrument inthe portfolio.
 15. The system of claim 14, wherein the processor issuesfurther instructions to: aggregate the simulated financial instrumentprices to produce a set of simulated portfolio value scenarios; andanalyze the simulated portfolio value scenarios to determine a value atrisk for the portfolio.
 16. The system of claim 13, wherein theprocessor issues further instructions to: symmetrize the set ofhistorical residual values prior to the use of selected historicalresidual values.
 17. The system of claim 13, wherein the processorissues further instructions to: increase the quantity of historicalresidual values available for use by applying a multi-day bootstrappingprocedure to the set of historical residual values.
 18. The system ofclaim 17, wherein a two-day bootstrapping procedure is used.
 19. Afinancial instrument portfolio attribute value simulation system,comprising: a memory; a processor disposed in communication with saidmemory, and configured to issue a plurality of processing instructionsstored in the memory, wherein the processor issues instructions to:provide an at least one parametric model for each of a plurality offinancial instrument attribute values associated with a financialinstrument portfolio, wherein each financial instrument attribute has atleast one noise-varying parameter α_(n) with a corresponding noisecomponent ε_(n); determine values for each of the at least oneparameters and the financial instrument attributes at various timeindices i using historical data; derive a set of historical residualvalues ε_(n,i) for each noise component ε_(n), the historical residualvalue ε_(n,i) at index i, when applied to the model with the determinedparameter values at index i, at least substantially reproducing thedetermined financial instrument attribute value at index i; standardizeeach set of historical residual values ε_(n); use values selected fromthe set of standardized historical residual values ε_(n) as the noisecomponent during a simulation of the financial instruments attributesvalues via the at least one model, wherein the processor issues furtherinstructions to use values for a plurality of scenarios for eachfinancial instrument in the portfolio to produce a corresponding set ofsimulated financial instrument attribute value scenarios for eachinstrument in the portfolio; aggregate the simulated financialinstrument attribute values to produce a set of simulated portfolioattribute value scenarios; and analyze the simulated portfolio attributevalue scenarios to determine an at least one attribute value at risk forthe portfolio.
 20. A financial instrument price simulation system,comprising: a memory; a processor disposed in communication with saidmemory, and configured to issue a plurality of processing instructionsstored in the memory, wherein the processor issues instructions to:provide a parametric pricing model having at least one parameter α_(n)with a corresponding noise component ε_(n); determine values for theparameters at various time indices i using historical data; derive a setof historical residual values ε_(n,i) for each noise component ε_(n),the historical residual value ε_(n,i) at index i, when applied to themodel with the determined parameter values at index i, at leastsubstantially reproducing the attribute price at index i; standardizeeach set of historical residual values ε_(n); apply a multi-daybootstrapping procedure to each set of historical residual values toincrease the quantity of historical residual values in each set; and usevalues selected from the set of historical residual values as the noisecomponent for the corresponding parameter during a simulation of theinstrument price.
 21. The system of claim 20, wherein the at least oneparameter comprises volatility.
 22. The system of claim 20, wherein theat least one parameter comprises drift.
 23. The system of claim 20,wherein the historical data includes implied drift.
 24. The system ofclaim 20, wherein the historical data includes volatility.